We will be using some JAVA applets to demonstrate the construction of position-time and position velocity diagrams.

The only quantities that you will be able to measure are:

Time

Length

In all cases the motion will be that of Motion Under Constant Acceleration

Some questions to think about are the following:

- What does the position-time plot qualitatively look like under
conditions of constant acceleration?
- What does constant acceleration mean in terms of forces
and velocities?
- What is the relation between distance and time in quantitative
terms?
- Why is friction analagous to gravitational free fall in terms of the
position-time plot?
- What does the velocity versus time plot look like under
conditions of constant acceleration?
- What is the relation between distance, time and acceleration, that
we can deduce from the data we just generated in class?
Let's being by considering the following diagram, which is a position vs time diagram describing the motion of some object under constant acceleration.

This diagram was made by using an appartus known as a fan-cart schematially shown below.

We can define the average velocity as the change in position divided by the change in time or:

Dp / Dt which is the slope of the line in the position vs time graph. By graphical means we then determine the velocity at any time by determining the slope on the graph at that time. This is what calculus is about but we aren't doing that here.

So, for instance, the average velocity of the cart over the time interval of 6 to 14 seconds would be equal to 1/4 meter per second (the cart moved 2 meters in 8 seconds).

## FORCES, ACCELERATION AND FRICTION

### The next kind of graph that can be made is a velocity-time graph. This is shown below:

Where we see that there are obvious changes in the velocity at various times. We are interested in understand what causes these changes in velocity.

In general an acceleration causes a change in velocity and thus we can define an average acceleartion in a manner that is similar to the definition of average velocity. In this case an average acceleration would be the change in velocity over some time interval or

Dv / Dt So, for example, over the period 6-8 seconds there was a change in velocity of 1.5 meters per second or an acceleration of:

1.5/2 = 0.75 meters per second

^{2}The fan attachment exerts an almost constant force on the cart. What is the relationship between that strength of that force and the cart's acceleration?

Possible hypothesis

- Acceleration depends on the total force. More force, more acceleration?
- Acceleration only depends on the initial force (the impulse) and not on a continuous force. Initial acceleration to constant velocity or constant acceleration?
- Acceleration depends on mass. More mass, more acceleration?
- Acceleration is inversely proportional to the mass. More mass, less acceleration?

We can qualitatively test this set of conjectures by considering the applet below. First set the friction slider to 0.

Then select a Force and Select a mass. Try to use different forces with the same mass or different masses with the same force. Observe the behavior of the position vs time and velocity vs time diagrams for different combinations of forces and masses. This should help you understand the relations between force, mass and acceleration.

Finally, turn friction to 0.1 and repeat the measurements. How does the velocity vs time diagram with friction = 0.1 compare to friction = 0.0. Is it linear or is it curved? If its curved, what do you think that means?

Definitions for speed, velocity, acceleration, force and mass in words.

- Average
**speed**of an object during a given time interval is the change in its position divided by the length of that time interval. is a measurement including both the*Velocity**speed*of and the*change in direction*of an object. To completely specify an*average velocity*, one must describe both its average speed and its average direction over a given time interval.*Average velocity*is the change in position-- including any changes in direction-- divided by the change in time. --In the simple case of motion along a straight line, average velocity would be the change in position divided by the time interval--.is a measurement of both the change in speed and the change in direction of an object. Average acceleration is the change in these two divided by the time interval. --For the case of motion along a straight line, average acceleration would be the change in velocity (which can be negative or positive) divided by the time interval--.*Acceleration*- Experiments show that the
**acceleration**of an object**is directly proportional****to the net force**acting on it. This has several ramifications:- As the net force applied on an object is increased, so is the object's acceleration. If the acceleration of an object is observed to decrease, then the net force acting on it must also have decreased in magnitude.
- The direction of the acceleration is the same as the direction of the net force.
- If no net force is exerted on an object, then its acceleration is zero.

- Experiments also show that, for a given net
force acting on an object,
**the acceleration of the object is inversely proportional to its mass**.- Thus, if the mass of an object is decreased, its acceleration will increase.
- Mass in this context is sometimes called
*inertia--*its resistant to change in motion. An object with greater inertia will resist change in motion (specifically, it will accelerate less) than a less massive object, all other things being equal.

- These two observations taken together can be
summed up as
**Newton's Second Law**:*The acceleration of an object is proportional to the net force acting on it, and the constant of proportionality is the object's mass.*